HIGHTEMPERATURE SUPERCONDUCTORS AS ELECTROMAGNETIC DEPLOYMENT AND SUPPORT STRUCTURES IN SPACECRAFT NASA NIAC Phase I Final Report MIT Space Systems Lab 10/1/2012
Gwendolyn V. Gettliffe Niraj K. Inamdar RESEARCH MENTOR: Dr. Rebecca Masterson PRINCIPAL INVESTIGATOR: Prof. David W. Miller
[email protected] [email protected] [email protected] [email protected] Electromagnetic deployment of space structures
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Table of Contents 1 Chapter 1  Introduction ........................................................................................................ 4 1.1 Motivation ................................................................................................................... 4 1.1.1. Reduced mass ...................................................................................................... 5 1.1.2. Larger structures with same launch vehicles....................................................... 7 1.1.3. Vibration and thermallyisolated structures ...................................................... 9 1.1.4. Staged deployment, inspace assembly, and partial system replacements...... 10 1.1.5. Reconfiguration of structures after deployment............................................... 10 1.2 Study objectives/Research questions ....................................................................... 11 2 Chapter 2 – Background....................................................................................................... 14 2.1 Scientific principles enabling HTS structures ............................................................ 14 2.1.1 Generation of Lorentz and Laplace forces ......................................................... 14 2.1.2 Meissner effect, superconductors, and manufacturing of HTS wire................. 15 2.1.3 Space environment ............................................................................................ 18 2.2 Enabling technology and previous work ................................................................... 19 2.3 Technology readiness ................................................................................................ 25 3 Chapter 3 – Theoretical Approach ....................................................................................... 26 3.1 Dynamics of unrestrained coils ................................................................................. 26 3.1.1 Rectilinear motion ............................................................................................. 26 3.1.2 Rigid body dynamics and rotation ..................................................................... 28 3.2 Incorporation of constraining elements.................................................................... 29 3.2.1 Tethers ............................................................................................................... 29 3.2.2 Hinges................................................................................................................. 36 3.3 Dynamic model implementation ............................................................................... 41 3.3.1 General solution algorithm ................................................................................ 41 3.3.2 Note on solution of stiff equations .................................................................... 42 3.4 Validation of numerical models ................................................................................ 43 3.4.1 Analytic dipole model ........................................................................................ 43 3.4.2 Sources of error in numerical approximation .................................................... 44 3.4.3 Validation models .............................................................................................. 46 3.4.4 Validation results ............................................................................................... 48 2
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3.4.5 Note on elasticity of panel ................................................................................. 56 3.4.6 Motion of a coil under its own force in Expand configuration .......................... 57 3.5 Conclusion ................................................................................................................. 60 4 Chapter 4 – Feasibility.......................................................................................................... 61 4.1 Electromagnetic structure design variables .............................................................. 61 4.2 Reformulation of equations of motion for trade space analysis .............................. 63 4.3 Trades ........................................................................................................................ 67 4.3.1 Six Degree of Freedom (6DoF) free ................................................................... 68 4.3.2 Tethered constraints in Separate configuration ................................................ 70 4.3.3 Hinged constraints in Unfold configuration....................................................... 83 4.4 Conclusion ................................................................................................................. 89 5 Chapter 5: Viability.............................................................................................................. 90 5.1 Alternative structural technology trades .................................................................. 93 5.1.1 Comparison of HTS structures to alternative structural technologies .............. 96 5.2 Internal variable trades ............................................................................................. 97 5.3 Phase II study plans ................................................................................................. 103 5.4 Summary and conclusions ....................................................................................... 104 6
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Works Cited ....................................................................................................................... 106
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Chapter 1  Introduction
1.1 Motivation This report, concluding a oneyear NIAC Phase I study, describes a new structural and mechanical technique aimed at reducing the mass and increasing the deployedtostowed length and volume ratios of spacecraft systems. This technique uses the magnetic fields generated by electrical current passing through coils of high‐temperature superconductors (HTSs) to support spacecraft structures and deploy them to operational configurations from their stowed positions inside a launch vehicle fairing. The chief limiting factor in accessing space today is the prohibitively large launch cost per unit mass of spacecraft. Therefore, the reduction of spacecraft mass has been a primary design driver for the last several decades. The traditional approach to the reduction of spacecraft mass is the optimization of actuators and structures to use the minimum amount of material required for support, deployment, and interconnection. Isogrid panels, composite honeycomb panels, and gasfilled inflatable beams all reduce the mass of material necessary to build a truss or panel, provide separation between elements, or otherwise apply surface forces to a spacecraft structure. An alternative to these “traditional” methods is the use of electromagnetic body forces generated by HTSs to reduce the need for material, load‐bearing support, and standoffs on spacecraft by maintaining spacing, stability, and position of elements with respect to one another. HTS structures present an opportunity for significant mass savings over traditional options, especially in larger systems that require massive structural components. Electromagnetic body forces generated by superconducting magnets are used to move and position spacecraft elements in lieu of traditional structural components, such as telescoping beams, segmented masts and inflatables. HTS structures have less mass per unit characteristic length of the spacecraft than aluminum beams and therefore offer the performance benefits of larger deployed structures while enabling the stowed structure to fit into existing launch vehicle payload fairings. However, the major cost of using HTS structures is the need to cool them to low temperatures so that they become superconducting, which requires passive cooling structures like heatshields or active cooling subsystems like cryogenic heat pipes. This work will also discuss the use of nonsuperconducting conductors for smaller forces or distances when passive cooling is not available. HTSs (which in general are superconducting at temperatures below 77K) and roomtemperature conductors can be utilized in tandem to perform more complex operations. This section explains the five primary benefits that HTS structures can offer to the aerospace community: 1. Reduced mass 2. Larger structures with same launch vehicles 3. Vibration and thermallyisolated structures
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4. Staged deployment, inspace assembly with fewer operations, and partial system replacements 5. Reconfiguration of structures after deployment 1.1.1. Reduced mass HTS structures can change the way mass scales with size; above some characteristic size for each spacecraft structure, the HTS system weighs less than other methods. For small, simple structures, the mass associated with single or multiple electromagnets and their power systems may be greater than that of traditional or other low mass structural options. In those cases, HTS structures may not be an advantageous choice (unless they present an additional benefit to the system; see benefits 35 below). For larger structures, however, the initial mass cost of electromagnets is small compared to the mass of a solid, rigid structure. After the payload, the structural subsystem is the second largest average fraction of spacecraft dry mass across all mission categories, as shown in Table 1, making structural mass the obvious target for mass reduction. [1] Table 1: Top 4 most massive subsystems average % of dry mass [1]
Subsystem (% of Dry Mass)
No Prop (%)
LEO Prop (%)
High Earth (%)
Planetary (%)
Payload
41%
31%
32%
15%
Struct/Mech
20%
27%
24%
25%
Power
19%
21%
17%
21%
Attitude D&C
8%
6%
6%
6%
One goal of our work, though not addressed more than qualitatively in the Phase I work, is to determine at what size HTS structural options become less massive than alternative structures. In order to compare the different alternatives, we define linear structural density as being the mass of the structural subsystem (including the mass of those elements of the electrical and thermal subsystems dedicated to structural deployment and support) divided by the length of the structure (for long, thin structures like booms) and areal structural density as being the mass of the structural subsystem divided by the area of the structure (for large, broad structures, like heatshields). In contrast, the term net areal density is used in this work to refer to the total mass of a spacecraft per unit area of a specific characteristic structure, the area of which dictates a key performance parameter of the mission, such as the primary mirror of a telescope. Examining the net areal density of space telescopes over time reveals an unsurprising trend over time towards larger mirrors with lower areal densities. This trend is shown in Figure 1.
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Figure 1: Chronological progression of net areal densities of space telescopes; mirror diameters listed beside points
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As shown in Figure 1, each successively newer mission has a lower net areal density than the last, and each, with the exception of Hubble, has a larger primary mirror than the last. The James Webb Space Telescope (JWST) is the first space telescope being built with a primary mirror larger than the diameter of its intended launch vehicle and, as such, the mirror must be hinged, folded, and then deployed from its stowed position within the launch vehicle. As the available launch vehicle fleet remains relatively static in fairing size over the foreseeable future, larger space structures will require additional, likely complex and operationally risky deployments in order to fit inside the fairing envelope, which suggests another potential benefit of HTS structures. 1.1.2. Larger structures with same launch vehicles Many space structures have performance benefits at larger sizes, but spacecraft size is limited by fairing envelope dimensions and maximum takeoff weight of launch vehicles. The high compaction ratio of HTS structures (related to how the stowed dimensions compare with the fullydeployed size of the spacecraft) means that spacecraft designers can reap the benefits of larger structures while being less sizelimited by the launch vehicle. For instance, a stack of HTS coils and tethers that can deploy to a boom length of tens of meters may be less than a meter in stowed stack height, whereas aluminum beams can only be as long as the maximum dimension of the fairing envelope before requiring hinges and actuators. Figure 2, from Bearden, 2001 [2] , shows how launch vehicle costs increase with both orbit altitude and satellite dry mass, and how spacecraft going beyond LEO cost twice as much per kilogram as those launching to LEO. If HTS structures can reduce the mass of a spacecraft by even 5%, the potential savings, both in cost and usable launch vehicle capacity, are enormous.
Figure 2: Launch vehicle cost with respect to orbit altitude and satellite dry mass. [2]
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Example performance benefits of bigger structures:
Larger primary telescope mirrors can observe objects farther away because of finer angular resolution and increased effective aperture.
The above equation shows the relationship between diffractionlimited angular resolution Θ, wavelength λ, and aperture diameter D for a circular aperture. Angular resolution in the telescope case is the minimum angle between distinguishable objects in the image produced by the telescope’s objective, or primary mirror, which serves as the aperture when using the above equation. The larger D is, the smaller sin Θ is, which for small angles means a smaller Θ, thus giving a finer angular resolution and the ability to distinguish smaller distances in the telescope’s image, unless the payload’s resolution is otherwise limited by the design of the focal plane sensor.
Larger solar sails provide more thrust via greater surface area over which solar pressure acts.
Larger parabolic radio frequency antennas have higher gain and can enable more distant missions or increased transmission data rates.
where is the antenna efficiency, is the wavelength of the signal being transmitted, and A is the effective antenna area. Gain varies directly with A.
Larger solar panels can hold more photovoltaic cells and thus generate more power. Longer synthetic aperture radar arrays achieve finest resolution in both cross and alongtrack dimensions by minimizing alongtrack (AT) dimension and maximizing crosstrack (CT) dimension. Larger heatshields can keep more (or bigger) equipment cold.
While larger structures provide performance benefits, launch vehicle constraints limit payload linear dimensions and mass. Table 2 shows the payload capacities of the largest currently available and proposed launch vehicles (sans the SLS, which has no hard specifications available at this date but is rumored to have an 8m diameter payload fairing), including the maximum mass of payload to LEO, the fairing diameter and usable envelope diameter within the fairing, and the maximum fairing height for the vehicle. Green rows are the current heavy launch fleet; blue are proposed or indevelopment vehicles; and brown are retired vehicles. 8
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Table 2: Current and proposed launch vehicle specifications
Large Launch Vehicles
Fairing diameter (m) / usable envelope (m)
Max fairing height (m)
Payload to LEO (kg)
Delta IV Heavy
5.1/4.57
9.72
23000
Ariane V
5.4/4.57
17
21000
Proton
4.35/4
15.25
20700
Atlas V HLV (proposed)
5.4/4.57
13
29400
Falcon Heavy (in dev)
5.2/4.6
11.4
53000 (~26000 if fully reusable)
Space Shuttle (retired)
~4.6
8
24400 after Challenger, 29000 was original specification
Structures larger than about 4.5m in two or more dimensions are thus unable to fit into any of the currently available launch vehicles without needing to be condensed in at least one dimension. HTS structures provide a means for increasing the deployedtostowed length ratio of a structure that can be launched by one of the current fleet of launch vehicles. 1.1.3. Vibration and thermallyisolated structures Structures that are electromagnetically supported (and thus have empty space between them instead of solid structures) minimize the conductive pathways between spacecraft elements. This enables better thermal and vibrational isolation for sensitive components than if other parts of the vehicle are physically joined together. The pointing accuracy and image quality of precision instruments is affected by coupled vibrations or jitter in the structure from internal or external disturbances (the latter including gravity gradient torqueing and solar pressure). Longer structures have a greater displacement at their far end than do shorter structures for the same angle rotation as well as a lower fundamental frequency, which is undesirable due to potential coupling with the launch vehicle during launch and with the attitude control system during operations. Attitude control of the spacecraft becomes much more difficult when the fundamental frequency is low.
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The cryocoolers used with cryogenic heatpipes for cooling superconducting coils generate their own disturbance profiles, so vibrational isolation is more approachable in systems already possessing passive areal cooling in the form of a sun or heatshield. 1.1.4. Staged deployment, inspace assembly, and partial system replacements In combination with electromagnetic formation flight, which will be discussed in greater detail in Section 2, HTS structures minimize or eliminate connections between spacecraft parts such that a heatshield and a mirror assembly of a spacecraft like JWST, for instance, do not have to be launched at the same time. Additionally, the entire spacecraft would not have to be replaced should one of the two fail because the parts are independent and do not require much or any physical assembly. This could give large, complex spacecraft longer lifetimes and the ability to be serviced or replaced at distances as great as EarthSun L2. 1.1.5. Reconfiguration of structures after deployment Reconfigurability is a spacecraft function that is not currently cost effective since two different functional designs on a single vehicle require a large amount of additional structural/mechanism mass, and the benefits are usually not worth the added design complexity. HTS structures allow for dynamic changes in boom length, solar array placement, and heatshield angle, as well as reversible deployments in some configurations – the capability to reconfigure could create a whole new paradigm of multipurpose spacecraft. Much of the analysis done in Chapter 4 supports the capabilities of HTS structures for use in Reconfigurability, and while HTS structures carry a significant power and thermal burden in their use, in large, highperformance systems, unique capabilities like Reconfigurability might be worth the trade, especially in designs where the coils do not have to be superconducting and in use at all times. This Phase I study and report aims to provide information about the potential of HTS structures for such unique functions as well as information about the feasibility of HTS structures overall.
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1.2 Study objectives/Research questions The three major questions that this report addresses are related to the technical feasibility of HTS structures, the spectrum of viable applications to spacecraft, and the new operations enabled or made substantially more realistic by HTS structures respectively. Question 1: Can we use electromagnetic forces generated by and acting between hightemperature superconductor currentcarrying coils to move, unfold, and support parts of a spacecraft from its stowed position? In order to answer that question, we first define functional configurations of coils that can be used to perform specific operations onboard a spacecraft, distinguished from each other by their degrees of freedom, starting and ending positions, number of coils, and whether they are done at initial deployment of a structure or at a point later in the operational lifetime. HTS coils can repel, attract, and even shear with respect to one another; in a flexible, noncircular coil, elements of the same coil can perform these actions on one another, deforming the shape of the coil over time. In combination with various boundary conditions, these operations lead to the seven functional coil configurations, depicted in Table 3, that we have identified for use in spacecraft deployment and support activities: four (14, in blue) are for initial deployment and three (57, in green) are variants of the deployment configurations for use in overall spacecraft shape change during the spacecraft operational lifetime. Table 3 describes the configurations and the spacecraft operations they could perform. These configurations will be capitalized when referred to in this report.
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Table 3: Seven potential functional coil configurations
Configuration 1. Expand
2. Inflate
3. Unfold
4. Separate
5. Deform
6. Reconfigure
7. Refocus
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Phase
Description
Uses
Deployment A single, flexible HTS coil that is folded in its stowed position and uses its own magnetic field when current is run through it to Expand to flat shape Deployment A 3D structure is built with two or more repelling coils in a configuration that creates a space between the two, Inflating a structure (bounded by flexible walls or tethers) Deployment A series of coils embedded in or attached to a structure that is stowed folded and must be Unfolded to become operational (folds can be hinges, springs, or couplings) Deployment Corollary of Inflate, in which two or more coils repel each other in series facing each other to Separate two parts of a spacecraft; tethered or membranous structure connecting Operational Two or more coils embedded in parts of the structure act magnetically on each other to temporarily Deform or change the shape of the spacecraft Operational Corollary of Deform, except Reconfigured state is sustainable and lasting
To deploy and hold taut the perimeters of large membranous or flexible structures
Operational
To change focus lengths of mirrors and gains of antennas by reforming or moving their dishes, mirrors, or horns to Refocus them on a new target
Two or more coils adjust their magnetic state such that an antenna or mirror is Deformed to Refocus it. Orientation of coils dependent on original shape of mirror or antenna.
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To create a volume (such as a tank or toroidal perimeter) or to increase surface area (for solar cells)
To deploy long, flat structures (like solar panels) or to take advantage of mechanical resistance at folds to create variable angles To put large, controllable distance between two sensitive parts of spacecraft (such as a nuclear reactor, astronauts, optics, thrusters) To reduce radar cross section (RCS) or adjust shape for avoidance of debris To reduce drag profile or Reconfigure satellite for different ConOps
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The four deployment configurations (14 in Table 3) are numerically modeled in MATLAB and Simulink to verify that the deployments can be performed as described in Table 3 and are validated in the farfield (where the coils are very far apart from one another) against the dipole approximation of coil forces and torques. Chapter 3 describes these models and how they were constructed and validated. Question 2: For which operations does this technology represent an improvement over existing or indevelopment options? In order to investigate the performance of HTS structures versus other lowmass structural technologies, the ranges of key performance parameters over which HTS structures are viable options for spacecraft design must first be determined. Chapter 4 characterizes the performance of HTS structures using the models introduced in Chapter 3, while in Chapter 5 they are compared to other structural options, both traditional rigid metal structures and “alternative” structures made with different materials and deployment operations. Question 3: What new mission capabilities does this technology enable? The Reconfigure, Deform, and Refocus operational configurations all represent mission capabilities that have heretofore required structures and mechanisms that are too expensive in metrics like mass, size, or power or are too complex and therefore too risky to justify for the additional performance benefits that they offer. This is why spacecraft do not commonly include these capabilities. Using magnetic forces, however, such shapechanging mission capabilities are not structurally much different from the deployment and support configurations required to move from a stowed configuration to a deployed one, so including the ability to Reconfigure or Deform utilizes the existing structural architecture more efficiently than in nonelectromagnetic structures. Operational configurations will be discussed further in Chapters 3 and 4. The goal of this report is to provide analyses of the dynamics of rigid (and, briefly, flexible) electromagnetic coils when exerting forces and torques on one another, such that we can support a positive answer for Question 1, narrow down the field of potential deployment and support functions that are worth further study for Question 2, and explore new spacecraft architectures that have not previously been worth seriously entertaining for Question 3.
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Chapter 2 – Background
This chapter introduces the foundational scientific principles for HTS and reviews previous development efforts to mature key enabling technologies for HTS electromagnetic structures. We discuss the enabling scientific principles and phenomena that form the basis of our study, draw lines between HTS structures and previous space electromagnetic work, and briefly examine the maturity of the HTS structure concept before moving into more theoretical analysis of the deployment dynamics in Chapter 3.
2.1 Scientific principles enabling HTS structures HTS structures are enabled by the fundamental scientific principles of electromagnetism and the unique environment of space, as well as the developments made over the last several decades in manufacturing superconductors and controlling electromagnets onboard multisystem vehicles. The industry development is explored in later sections; this section describes the underlying fundamental scientific principles that enable HTS structures, including: 1. The creation of Lorentz and Laplace forces via interaction of a magnetic field and current 2. The Meissner effect, superconductivity, and the manufacturing of HTS wire 3. Enabling characteristics of space environment (microgravity and vacuum) 2.1.1 Generation of Lorentz and Laplace forces HTS structures operate using electromagnetic forces to push, pull, and move with respect to each other. Electromagnetic forces (called Lorentz or Laplace forces depending on whether the force is acting on a single charge or a current of charges respectively) result from the interaction of a magnetic field and a current. A point charge q moving with velocity ⃗ in external magnetic and electric fields, ⃗⃗ and ⃗⃗ respectively, experiences a Lorentz force ⃗ , given by:
⃗
⃗⃗
⃗
⃗⃗
where q ⃗⃗ is the electric force, and q ⃗ ⃗⃗ is the magnetic force. The macroscopic force on a wire is the magnetic force and is called the Laplace force. It is generated by a magnetic field ⃗⃗ on a wire carrying current ⃗ (a stream of point charges) as follows: ⃗ 14
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⃗
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The magnetic force is thus orthogonal to the wire and to the orientation of the magnetic field at the point of calculation. In order to calculate the magnetic field ⃗⃗ for use in determining the Laplace force on a wire or coil of wire, one can use the BiotSavart law, which can be derived from Ampère's law and Gauss’s law, to compute the resultant magnetic field vector ⃗⃗ at a position r with respect to a steady current I: ⃗⃗
∫
⃗
⃗
 ⃗
In the deployment modeling work that will be described in Chapter 3, the magnetic field and the Laplace forces across a currentcarrying wire are approximated over time by implementing the BiotSavart law numerically, discretizing electric current elements in order to determine the magnetic field generated by arbitrary configurations of rigid (meaning a fixed, nonchanging shape) and flexible coils. Knowledge of the magnetic field at each point in space around a currentcarrying wire allows the calculation of the resultant force upon another currentcarrying wire as a result of that magnetic field, which can then be used to determine the number of turns a coil requires or how much current it needs to carry in order to deploy a structure.
2.1.2 Meissner effect, superconductors, and manufacturing of HTS wire Superconductors enable HTS structures because they are able to generate much larger forces via their larger current carrying capacity, which increases the distance over which they can work for the same amount of mass. Superconductors are materials that conduct electrical current perfectly below a critical temperature ; superconductors have zero resistivity, with negligible quantities right around their . Any current through them will persist significantly longer than through a nonsuperconductive material. Superconductivity is characterized by the Meissner effect, the expulsion of an external magnetic field from a superconductor once cooled below its during its transition to a superconducting state. Every superconductor has a critical temperature, external magnetic field strength, and current density above which superconductivity ceases, shown graphically in Figure 3.
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Figure 3: Critical surface for Type II superconductor [3]
Hightemperature superconductors, or HTSs, are those superconductors with s above 77K, or the boiling point of liquid nitrogen, enabling them to be cooled to a superconducting state using LN2. The higher the , the less it costs (in terms of power, storage, and consumables, for applications where a cryogen is not recycled) to do the cooling. HTS development and the subsequent development of HTS wire has led to a broad array of applications for superconductors, including long distance power transfer, electromagnets, and energy storage. There are two types of superconductors; TypeI only exhibit the Meissner effect with one critical field strength above which superconductivity ceases. TypeII superconductors, which include all hightemperature superconductors, as well as some lowtemperature superconductors (LTSs) with s too low to qualify as “hightemperature”, also exhibit a “mixedstate” Meissner effect that increases their critical magnetic fields and configuration stability. Because of this effect, TypeII superconductors are often used in superconducting magnets in the form of coils of wire made with superconductor filaments embedded in support material less than a millimeter in diameter. The “mixed” Meissner effect is different from the Meissner effect in that some magnetic field penetrates the superconductor through filaments of normalstate material, and the material can support higher magnetic fields before superconductivity breaks down. There are thus two critical field strengths in Type II superconductors: beyond the first field strength, where superconductivity would cease completely in a Type I superconductor, a vortex (“mixed”) state exists in which some magnetic flux is allowed to penetrate the material while it continues superconducting. Beyond the second, higher critical field strength, superconductivity ceases. TypeII superconductors are the only type of superconductor used in wire. Many of these types of wire are made with HTSs to lower cooling costs, especially for nonmagnetic applications. Some wire, however, especially that which is used in powerful electromagnets like those in the Large Hadron Collider, is made with LTSs that need to be kept much colder but 16
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can sustain much higher current densities than HTSs. For example, niobium–tin (Nb3Sn) has a of 18.3K and can withstand magnetic field strengths up to 30 tesla (with a record current density of 2643 A/mm^2 at 12T and 4.2K) [4]. HTS wires do not have the same current densities, and, as a result, they cannot generate magnetic fields as high as LTSs. But, though pure HTS materials are just as brittle as LTS materials, HTSs can be constructed into flexible, durable wire. The flexibility of HTS wire enables the Expand configuration that we proposed earlier in this section for deployment of a single, folded and stowed coil into a large, flat, expanded coil. Compared to standard, roomtemperature conductors, HTS wires are able to create larger magnetic fields and sustain higher current densities, with littletono resistive losses through the wire (compared to high resistive losses in copper and aluminum). While room temperature conductor coils can be used to magnetically operate on each other, the Laplace forces able to be generated on each other are significantly lower than those that can be created with HTS coils, due to the lower induced magnetic field. HTS wire thus enables electromagnetic structures with multimeter separation between coils, which in turn enables larger vehicles and the performance benefits enumerated in Chapter 1 that accompany larger structures. Whereas several decades ago it was difficult for manufacturers to create HTS wire of any useful length, HTS can now be created in lengths upwards of 1500m [5], reducing the need for splices that result in resistive losses of power to heat. Individual strands of tapeshaped (wide and thin) cable are multilayered with various substrates and insulators for mechanical strength and chemical stability while at the same time trying to maximize the current density of the wire. Figure 4 shows a cutaway view of an HTS wire.
Figure 4: Cutaway view of a 2G HTS wire [6]
Roebel cables, shown in Figure 5, implement a way of winding together individual wire strands to reduce AC losses due to selffield interaction with the current running through each strand, reducing the critical current able to pass through that wire. When determining the configuration of turns in a coil or strands in a cable, it is especially important to consider how the selffield affects the in each strand to optimize for the maximum current density in the coil crosssection possible. 17
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Figure 5: 15strand HTS Roebel cable [7]
SuperPower Inc, a subsidiary of Furukawa Electric Co., Ltd, manufactures a flexible “2G”, or second generation, HTS wire with a minimum bend diameter of 11mm at room temperature (and at least 25.4 mm at superconducting temperatures) and maximum hoop stress of 700 MPa before irreversible degradation of the critical current. Recent axial stressstrain measurements conducted at the Naval Research Laboratory on standard 4 mm wide SCS4050 production wire gave a yield stress for the wire of 970 MPa at 0.92% strain. [8] The 2G HTS wire can carry between 250 and 350 A/cm wire width (or 25 – 35 kA/cm2 with a .1mm wire thickness) through 650m lengths (80110A critical current with a 4mm wire width) [6]. We use the properties of this 2G wire in our models. 2.1.3 Space environment On the ground, gravity and the need to cool superconductors in an otherwise roomtemperature environment makes using electromagnets as actuators or structural support difficult and not broadly useful. In the space environment, however, an electromagnet does not need an enormous magnetic field to actuate components, and there is no air transferring heat into the magnet by convection, making space a potentially favorable environment for the use of HTS structures. The microgravity environment of space enables HTS structures because spacecraft elements can be actuated without overcoming gravitational forces. Thus, only small forces are needed to cause motion or actuation, reducing the necessary size and current of electromagnets used for such tasks compared to what would be needed, for instance, to repel a coil upwards on the Earth’s surface.
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The vacuum of space is both beneficial and detrimental to the use of superconductors. If it can avoid radiated input from the sun, Earth, or the space vehicle, a thermally isolated HTS subsystem can maintain superconducting temperatures without cryogens. However, it is difficult to remove heat from the system in a vacuum if it cannot be isolated from conductive or radiating heat sources. This thermal environment makes more advanced cooling systems, like the cryogenic heatpipe described in Section 2.2, necessary for maintenance of the superconductor below its critical transition temperature.
2.2 Enabling technology and previous work A number of previous and ongoing industries and projects lend credibility to the concept of using HTS to actuate spacecraft elements with respect to one another, including (1) dynamics and control technologies and testbeds like MIT SSL’s electromagnetic formation flight (EMFF) testbed and the University of Maryland’s (UMD) RINGS project (slated to fly to ISS in December 2012) Peck’s flux pinning [9], and Pedreiro’s Disturbance Free Payload [10]; and (2) thermal control innovations like Sedwick’s cryogenic heatpipe for HTS coil cooling. 1. Dynamics and control of a spacecraft without a support structure become a significant challenge and represent the majority of the work that must be done to design a system of electromagnets, since actuation and support are dictated by the feedbackinformed distribution and direction of power through wire. The MIT SSL and UMD Space Power and Propulsion Laboratory have studied the problem of controlling electromagnetic coils in 3 and 6DOF systems for the last decade, showing that control of freeflying or tethered vehicles with electromagnets is feasible and reinforcing the concept of using HTS coils to deploy a structure. Electromagnetic formation flight (EMFF), has been the subject of study by the MIT SSL since 2002, funded by the original NIAC program (Phase I and Phase II studies), NRO DII, the JPL TPF Program, NASA GSFC SBIR, and DARPA. In 2003, the MIT AeroAstro senior capstone design class focused on the development of an algorithm for formation control using the intervehicle forces from steerable electromagnetic dipoles. As part of the class, and with funding from the NRO DII and the NASA Institute for Advanced Concepts, the MIT group implemented a laboratory testbed that used high temperature superconducting (HTS) wire to create electromagnetic coils. In a followon effort, MIT and Aurora Flight Sciences (AFS) investigated the performance of a roomtemperature conductor version of EMFF (termed microEMFF). This effort included both a performance assessment and construction of a groundbased microEMFF testbed. Resonant Inductive Nearfield Generation System (RINGS), is a joint UMD SPPL/MIT SSL/AFS project that, among other things, is a microEMFF demonstration onboard the International Space Station. These programs have included intensive control algorithm development as well as TRL development of the HTS EMFF concept to TRL 4 and the microEMFF concept to TRL 5, once RINGS begins operations on ISS. The basic concept of EMFF is to provide actuation in relative degrees of freedom for formation flight systems using electromagnetic forces/torques and reaction wheels. The positions of system elements can change so long as the center of mass of such a system remains fixed (barring external input). The motivation for EMFF in multivehicle systems is fourfold: 19
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Stationkeeping for distributed satellite systems Replacement of consumables (thrusters) Elimination of thruster plumes Enabling of high deltaV formation flying missions
Figure 6: MIT SSL EMFF Testbed [11]
In the 3DOF EMFF testbed built at MIT SSL, shown with subsystem labels and with both test vehicles in Figure 6, vehicles exert electromagnetic forces and torques on each other using two perpendicular HTS coil electromagnets each. A 6DOF system would use three orthogonal HTS coils to create a completely electromagnetically steerable magnetic dipole. The reaction wheel decouples torques, allowing independent control of a vehicle’s rotation. During MIT SSL’s previous work with this experimental testbed, several control algorithms were developed and validated. These results show that the relative position of spacecraft can be controlled and stabilized using 1 to 3 electromagnetic coils and 3 reaction wheels per vehicle [12]. While this particular testbed was not a consumablefree, closedloop system (LN2 cooled the superconductors but was allowed to boil off), the cryogenic heatpipe previously discussed enabled a closedloop system with no LN2 loss. RINGS, or Resonant Inductive Nearfield Generation System, is an electromagnetic formation flight and wireless power/data transfer testbed that will begin operations on the ISS in December 2012. It is an addon to the SPHERES (Synchronized Position Hold Engage and Reorient Experimental Satellites) developed by MIT SSL and is currently being operated by NASA Ames Research Center as a National Lab onboard the ISS. RINGS is an advancement of the microEMFF concept in that it uses roomtemperature conductors instead of 20
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superconductors, and its use on the ISS will advance the TRL of microEMFF to 5 via brassboard demonstration in a relevant environment. The RINGS consist of two 155turn, 0.685m outside diameter coils of noninsulated aircooled aluminum cable enclosed in a protective plastic shielding. One coil is mounted on each of two SPHERES as shown in Figure 7 and, using the SPHERES’ CO2 thrusters to steer the dipole, the coils can be used to maneuver with respect to one another magnetically.
Figure 7: A SPHERE outfitted with RINGS hardware in laboratory
The primary differences between the application of HTSs in EMFF and in structures/deployment are (1) the presence or absence of physical connection paths between interacting coils and (2) the level of realtime direction the system has over the movement of coils. EMFF vehicles can steer their dipoles in response to realtime system input to maneuver with respect to one another and have reaction wheels to cancel out torque when shearing with another vehicle. HTS structures are limited by their physical connection to the elements that they are moving, and their movements are planned for and constrained to certain, predictable paths, as is desired when deploying spacecraft elements to operational configurations. It is important to note that HTS structures can isolate different parts of a spacecraft vibrationally to a greater extent than can solid structures, but unless the elements are formation flying, there are still some transmission paths of forces and vibration, limiting movement of a coil in at least one degree of freedom. The EMFF project did not focus on deployment dynamics and the stability of transient states from one position to another without capacity for 6DOF control and steering of the resultant dipoles, thus further work is required to characterize this feasibility risk (and control design challenge) for the HTS structures application. Although controllability is a major component of technical feasibility for HTS structures, it is not the focus of this report. Electromagnetic actuation on spacecraft has been the subject of research by a number of groups because of its potential for vibrational isolation and reconfiguration. Disturbance Free Payload, or DFP, is an architecture developed by Pedreiro et al for use in vibration isolation of 21
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spacecraft payloads from the rest of the spacecraft using multiple electromagnetic actuators [10]. Flux pinning is an electromagnetic phenomenon in superconductors that enables highly accurate positioning between an electromagnet and a permanent magnet or other magnetic field source that can be used for safer and easier inspace assembly and reconfiguration of space systems in orbit. Small defects in the superconducting material can increase critical magnetic fields and contribute to the stability of the system by fixing vortex points, or filaments, through which magnetic field lines pass and are subsequently pinned when the superconductor is cooled below its . Pinning of the field lines pins the source of the magnetic field in position and orientation. A number of joints, hinges, and interfaces can be created using configurations of permanent magnets to limit or direct the motion of a superconducting cube about the permanent magnets. As of 2009, multicentimeter gaps between the magnets were supportable with high stiffness and damping with small masses. Flux pinning as a means of supporting and manipulating space structures is the subject of work by Peck et al at Cornell University [9]. Stiffening of membranous or low net areal density structures has been proposed on multiple occasions, such as Zubrin’s “magsail” concept in which a large membranous plasma wind sail is held taut using a flexible HTS coiled around its perimeter to repel itself into a circle and keep the sail under tension. Zubrin ultimately selected a nonmagnetic deployment system (rotating booms to deploy the sail initially using centrifugal force), citing “reliable deployment” as a key issue for magsails. [13] A previous NIAC study performed by Powell et al [14] proposed a system like that being studied in this report: flexible cables made of hightemperature superconducting wire Expanding or Inflating in order to serve as deployment actuators, perimeter support, and standoff structures simultaneously for large scale spacecraft. Powell primarily discusses the HTS wire properties and materials and does not go into detail on the deployment process. This report will explore such magnetic deployment in detail in an effort to determine if magnetic deployment is feasible, and if so, for what types and sizes of structures. Other past studies of deployment or support using electromagnetism in membranous structures include microwave beam‐driven spin deployment of solar sails (Benford, [15]) and membranous structures containing conductive meshes that can be shaped using magnetic pressure from permanent magnets or electromagnets (Amboss, [16]). For applications that require long, lowmass structures, HTS structures are more suited than flux pinning or DMP to deployment and support in those situations. The possibility remains, however, of using a combination of flux pinning, DMP, EMFF, and HTS structures to magnetically assemble, deploy, and support large and complex space structures. Other light structural technologies like inflatables or tensegrity structures do not have active control over their shape; they attempt to deploy and either succeed or fail, and then they passively maintain their configuration for the lifetime of the vehicle. The ability to control and change the magnetic field means that electromagnetic structures are able to change their shape after deploying, depending on their boundary conditions. 22
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2. Thermal control down to cryogenic temperatures is necessary to induce superconductivity in HTSs. This level of thermal control is not normally present on spacecraft unless the vehicle has a payload that requires special cooling, making the need for cooling one of the primary disadvantages of HTS structures. Additionally, HTS performance is dictated by its warmest temperature, meaning that cooling devices are required at every point along the coil, which is not feasible (or even accomplishable), for large coils. Cooling equipment that can cool the full perimeter of a coil is needed and must be scalable to different size coils. Such equipment should preferably be consumablefree so as not to limit the lifetime of the mission. If the thermal control requires too much mass and power, the thermal subsystem could make HTS structures less competitive with traditional structures by nullifying any potential savings of HTS structures. The cryogenic heatpipe developed by Sedwick and Kwon [11] for use in EMFF, pictured in Figure 8, provides a starting point for continued development and improvement of a competitive HTS coil cooling system by accomplishing consumablefree isothermalization to cryogenic temperatures (the same temperature at every point on the coil) with a single cryocooler.
Figure 8: Crosssection of heat pipe (left) and cryogenic heatpipe (copper casing) in open toroidal vacuum chamber (right)
This rigid heatpipe uses nitrogen as a working fluid and a stainless steel mesh as the wicking structure, with the HTS coil enclosed within the heatpipe instead of residing externally as with traditional heatpipes. A condenser, operated with an LN2 reservoir (like in the laboratory setup shown in Figure 8) or powered by a cryocooler, extracts heat from the system while the mesh passively wicks the condensed LN2 around the pipe via capillary action on several layers of wire mesh to cool and achieve isothermalization of the coil. Isothermalization means that minimal temperature gradients exist across the entire coil. Isothermal conditions are desired because the warmest point in the coil is the limiting factor in the performance of the HTS wire. When used with a cryocooler, as in the composite heatpipe design discussed next, this heatpipe is a closed system wherein the working fluid is conserved so long as the system remains sealed.
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UMD SPPL conducted additional heatpipe development under a NASA STTR, in cooperation with Axis Engineering Technologies, Inc. [17]. The principal heatpipe design remained the same; however, the focus was on manufacturability of the heatpipe, which was previously found to be problematic. The issue was that the superconducting coil had to be a continuous spiral, requiring that the housing be formed around the coil. The housing must also be thermally conductive but ideally electrically insulating, since eddy currents developed in the housing would generate heat and dissipate power. The resulting design is shown in Figure 9. The clamshell design is constructed from a thinwalled composite material, instead of the copper used in the first heatpipe, held together with a set of inner and outer clamp rings. The composite material is electrically insulating, but the thin wall allows for good heat conducting radially. The outside of the pipe is subsequently coated with MLI.
Figure 9: Twopart composite heatpipe showing an HTS coil inside
The cryogenic heatpipe, about two meters in diameter with 100 turns of HTS coil within its 4cm pipe diameter, was designed with a heat capacity of 100W (determined from the measured radiated heat input from the vacuum chamber walls) and succeeded in keeping the HTS coil below its critical temperature of 110K. The mass of the 2m diameter copper pipe was approximately 70.5 grams. In LEO, however, the worstcase heat capacity of a 1 meter diameter MLIcoated coil (the size used in the EMFF testbed) is 5W [11]. 5W of heat can be extracted using a relatively lowmass (2.1 kg) cryocooler like the Sunpower CryoTel MT [18].
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2.3 Technology readiness HTS structures are a logical evolution of the previously realized or studied applications of electromagnets in space, and they bridge the gap between electromagnetic actuation, support, and positioning as a system that is capable of all three functions. In the future, spacecraft designers can tie together many of these positioning and actuation applications into a single or collection of multiple spacecraft, utilizing formation flying, easy insitu assembly, isolation and high stowedtodeployed size ratio structures to enable spacecraft that have never before been possible. Therefore, in this report, we will address several of the questions relevant to the feasibility and viability of implementing HTS structures through modeling and trade analysis. Furthermore, a goal of this work is to mature the technology readiness level (TRL) of HTS structures from conceptualization and early analysis of this application (TRL 2) to a more detailed analysis and hardware proofofconcepts (TRL 3). Phase I encompasses the transition from TRL 2 to early 3 while our Phase II work aims to mature the technology further through TRL 3.
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Chapter 3 – Theoretical Approach
This chapter will detail the equations of motion that describe the dynamical evolution of currentcarrying coils in space and how this motion was simulated using MATLAB and Simulink models. In this report, we will be primarily discussing the motion of two rigid coils under a variety of boundary constraints; we will briefly discuss the motion of flexible coils in this chapter with a more indepth discussion and analysis reserved for our Phase II report. These particular models were selected because between the two of them, all seven functional configurations presented in Chapter 1 could be modeled given appropriate boundary conditions. In general, any currentcarrying coil will experience forces and torques due to other currentcarrying coils and magnetic fields in the vicinity. One of the purposes of this work is to determine how the mutual evolution of a network of coils is dependent upon the initial positions and orientations of the coils and how that evolution can be controlled for deployment and subsequent support purposes by imposing various constraints upon them, such as flexible tethers and hinges. Accurate models of coil dynamics can be used to characterize the range of system parameters over which electromagnetic structures are potentially useful. In this chapter, these models and the methods used in their construction will be introduced for use in studying the ranges over which each of the functional configurations is feasible.
3.1 Dynamics of unrestrained coils The coil dynamics can be described fully by Newton’s laws of motion. Newton’s 2nd law will be employed in both its rectilinear form, for modeling the motion of the center of mass of the coil, and in its angular form, in order to describe the rigid body rotation of the coils through space. Newton’s laws are first supplemented with the appropriate electromagnetic laws introduced in Chapter 2 to describe the electromagnetic forces and torques exerted on each coil by one another in a six degreeoffreedom model and later with mechanical forces that reflect the influence of constraining devices (tethers and hinges) on the coils. 3.1.1 Rectilinear motion Consider a set of current loops, one of which we label by the subscript i, where i = 1,2,…, N, with N being the number of coils in our system. The magnitude of the current in coil i is given by At some point P in space (defined with respect to a suitablychosen inertial coordinate system), the magnetic field due to a directed, differential coil element ⃗ is given by Ampere’s law: ⃗⃗
⃗
⃗
⃗ 
where ⃗ ⃗ ⃗ is the vector from P to the differential coil element, ⃗ is the vector between two infinitesimally close points on the current loop, and is the magnetic constant, equal to T·m /A. 26
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The total magnetic field at P due to coil i is given by the integral over the entire loop, or: ⃗⃗
⃗
∮
⃗
⃗ 
Loop
Now, suppose P happens to be on another current loop (denoted “j”), with the magnitude of its current given by . Let the differential coil element at P be given by ⃗ ; then, by the Lorentz law, the force exerted on this differential coil element by the magnetic field due to coil i is given by: ⃗
⃗
⃗⃗
⃗
⃗
∮
⃗
⃗ 
Loop
where ⃗ has been rewritten from ⃗ to emphasize that the magnetic field from coil i is acting to create a force on the differential coil element ⃗ . and are constant values around the current loop, and in cases in which each coil is made up of wire turned multiple times, if is the number of turns in the ith coil, we may define an effective current as follows:
The total force exerted on coil j by coil i is then given by integrating the Lorentz force over coil j: ⃗
⃗
∮ Loop
∮ Loop
⃗
⃗
⃗ 
By Newton’s 2nd and 3rd laws, the force ⃗ acting on coil i from coil j is given by ⃗
⃗
⃗,
these forces acting on the centers of mass of each coil. For a system with more than two coils, the force is equal to the summation of the contributions from each of the N1 other coils (N1 instead of N because the selfcontribution of force from coil j is 0). Since a coil responding to the magnetic fields of other coils will move and rotate, the vector variables of current element direction and position in the above equation change with time for each coil in the system. As such, the delay in the response of a given coil due to the finite propagation speed of electromagnetic waves must be taken into account. In our case, however, so long as we are interested in time scales much greater than  ⃗ ⁄ , where c is the speed of light (3×108 m/s), the effect of the delay will be negligible.
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By denoting the position of the center of mass of the jth coil in the inertial coordinate system by an overbar, ̅ , we have:
̅
⃗
∑
∮
⃗
Loop
⃗
∮
⃗
⃗ 
Loop
There is no clean analytical solution for this problem besides the above double integral. For an axially symmetric coil, however, the magnetic field at a point P along the axis of symmetry out of the plane of the coil has the analytical solution:  ⃗⃗ 
⁄
where ⃗⃗ is calculated in Tesla, R is the coil radius in meters, I is the current in amps, n is the number of turns of wire in the coil, and x is the distance in meters along the axis from the center of the coil. This solution is useful for calculating the force on a single particle along the coil’s axis, but not for the force at any other point. Therefore, a numerical approximation of the force between the coils is required in order to study the behavior of the coils over time, accomplished by numerically performing the line integrals above. In this report, we investigate the dynamics of the coils using numerical models constructed in MATLAB and Simulink.
3.1.2 Rigid body dynamics and rotation In addition to the translational, rectilinear motion of two coils in space, the torque and resultant rotation induced by asymmetric forces on differential elements across the coil must be taken into account. By the force expressions given above, the torque exerted on the ith coil about its center of mass is given by: ⃗
∑
∮ ⃗ Loop
̅
⃗
∮ Loop
⃗
⃗
⃗ 
where ⃗ is the location of the differential element ⃗ with respect to the global inertial coordinate system, and ⃗ ̅ is the vector from differential element ⃗ to the center of the coil, providing the moment arm of the torque formulation. 28
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As with any analysis of rigid body motion, care must be taken in how the orientation of each body (in this case, the individual coils) is parameterized. In this report, the orientation of the coils in space is parameterized by means of a quaternion. The use of quaternions to describe the orientation of a body in space is well established—for a full account of the theory, the reader is referred to, e.g. [19]. We describe the approach in relation to coil dynamics briefly: We attach a coordinate system (called the body coordinate system and denoted ) to the ith coil. The orientation of this coordinate system is encoded within the quaternion ⃗ via the direction cosines matrix , which specifies the coordinates of with respect to a suitably chosen global inertial coordinate system. The matrix , when multiplied to the right by the set of vectors that define the positions of the points around the coil around its center, effects the appropriate rotation upon the coil. The dynamical response of the coils is calculated using Newton’s 2nd law in angular form, or ⃗
⃗
Here, is the moment of inertia matrix of the ith coil calculated with respect to its center of mass, given by:
[
]
so that it is aligned with the principal axes of the coil; the third entry in We have chosen the diagonal of corresponds with the axis normal to the plane of the coil. The angular acceleration of the body is ⃗ . From this, the evolution of the quaternion—and the orientation of the coil—can be advanced accordingly. [19]
3.2 Incorporation of constraining elements The equations of motion presented above are for coils in empty space. Ultimately, mechanical elements (possibly in conjunction with appropriate current controllers for the coils) must be employed in order to execute desired geometrical configurations of the coils in space. The coils and these constraining elements will then serve as the primary structure of the spacecraft. Below, we describe incorporation of such constraints into the equations of motion. 3.2.1 Tethers 29
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For the Separate functional configuration, the use of tethers to connect magnetic coils to each other and restrict their motion eliminates the need to control currents in order to keep coils positioned the desired distances from one another, as they would have to be in electromagnetic formation flight. Tethers also reduce the risk associated with two or more coils accelerating away from each other during deployment. This risk reduction is performed via the tethers providing a corrective “spring” force for any overshoot of the desired separation distance. In this report, three tethers equally spaced around the perimeter of the coils are used to restrict the motion of the coils, as shown in Figure 10. In future work, we will consider more complex tether configurations.
Figure 10: Diagram of 3tether spacing between two coils
Three tethers are used instead of two because two tethers do not restrict the rotation of the coil around the axis drawn through the tether attachments (visualized in Figure 11), which could lead to a 180 degree flip of one or both coils once the tethers reach their full length during deployment.
Figure 11: Diagram of 2tether spacing and unrestricted motion
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In the Separate deployment, the coils start close to one another with tethers slack, as shown in Figure 12. Suppose the coils are aligned along their central axes, and the tethers—all identical to one another—are parallel to these axes. Once current is run through the coils in opposing directions, the coils begin to accelerate away from each other until the coils are apart, where is the unstretched length of a tether. If the tethers are stretched to a length , an elastic force due to the tension of the tethers is effected. If we define , then the ⃗ elastic force of a single tether on a given coil is   , where is the spring constant of the tether with the force directed towards the other coil. If there are three equal length and material tethers, then the total elastic force acting on a coil is  ⃗  . In this model, the tethers are not rigid and do not provide compressive forces for distances shorter than .
Figure 12: Three coil Separate configuration with slack tethers
Thus, in a two coil system with three identical tethers, when < , the forces acting on the coils are solely to due to electromagnetic interaction, while, when , the forces acting on the coils will be due to both electromagnetic interactions and elasticity, the magnitude of the elastic force given by . In the more general case of coils and tethers not aligned, the effect of the tethers on the coils may be determined as follows. Let the three tethers be indexed by α = 1,2,3. Each tether has a natural length beyond which, when stretched, a restorative elastic force results. If the length of the tether is less than this, the tether will slacken and not exert any force in the opposite direction. Let the tethers have elastic constants given by . If the distance between the endpoints of the tether is given by , the magnitude of the elastic force due to tether α is given by: ⃗  31
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where
and { To determine the directions in which each force acts, we consider the case in which the tethers are between only two coils. The tethers are attached to individual points on the coils. For tether α, let the vector describing the point of attachment on coil 1 be given by ⃗⃗ , while the point of attachment on coil 2 is given by ⃗⃗ . Thus, it is clear that:
 ⃗⃗ while the unit vector ̂
⃗⃗

parallel to the elastic force due to tether α is given by:
̂
⃗⃗
⃗⃗
 ⃗⃗
⃗⃗

With these definitions, we can write
⃗ ⃗
̂ ̂
where ⃗ is the force due to tether α exerted on coil 1 and ⃗ exerted on coil 2. The total force on each coil is then simply
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⃗
∑ ⃗
⃗
⃗
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is the force due to tether α
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Now, the tethers will also effect torques upon the coils. If we wish to determine the torques about centers of mass of each coil due to tether α, we simply take the moment of the elastic force with respect to the coil center of mass:
⃗
[( ⃗⃗
̅)
̂
]
⃗
[( ⃗⃗
̅)
̂
]
Thus, the total torque exerted by the tethers on each coil is given by
⃗
∑⃗
⃗
∑⃗
One concern with this deployment strategy is that oscillations of the coils will occur around the equilibrium length of the tether. Potential solutions to avoid or damp out oscillations include a current controller that reduces and even switches the direction of current in the coils such that the coils experience an attractive force (i.e. a force in the opposite direction of their deployment motion) and are thus slowed to a stop as they reach the equilibrium length . In this report, we do not explore a controlled method of damping in detail. Because any realistic tether will have intrinsic to it some damping due to friction between its constitutive materials. To account for this most generally, we would need to solve, simultaneously with those of the coils, the equations of motion of the tethers themselves. However, this is computationally intensive, requiring the additional solution of the three coupled partial differential equations (complete with tethercoil boundary conditions) of the elastic behavior of the tethers. For simplicity, therefore, we use instead a model in which the damping forces exerted on the coils are a function of the velocities of the tether attachment points only, as in a classical massdamper system. Thus, we consider the relative velocity ⃗⃗ of the two coil attachment points for tether α with respect to the inertial coordinate system:
⃗⃗
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[ ⃗⃗
⃗⃗
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The damping force is supposed proportional to this, the proportionality constant taken to be . It can be shown, then, that:
⃗
 ⃗⃗ 
[
⃗
 ⃗⃗ 
[
( ⃗⃗ ) √ ( ⃗⃗ ) √
̂
]̂
̂
]̂
The total damping force acting on the coils is given by:
∑ ⃗
The torques about the coil centers of mass due to damping from tether α are:
⃗
 ⃗⃗ 
[
⃗
 ⃗⃗ 
[
( ⃗⃗ ) √ ( ⃗⃗ ) √
̂
] [( ⃗⃗
̅)
̂
]
̂
] [( ⃗⃗
̅)
̂
]
where the B denotes that the torques are taken with respect to the center of mass of the coil body. The total damping torques given by:
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⃗
∑⃗
⃗
∑⃗
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Other options besides simple cable tethers attached at each end to a coil exist for the Separate deployment. Thin, rigid members with joints that lock open as the coils move apart in order to guarantee a fixed distance between the coils and reduce oscillations during deployment are also a possibility, though such constraints would be more massive and complex than simple tethers and thus less competitive in a design trade where mass is a driving parameter. The Separate coil configuration as modeled here is both a deployment and a support device; with rigid, locking members, it becomes purely a deployment device and thus loses the novelty of being able to perform two functions simultaneously. Another option, one that allows for Reconfiguration over the lifetime of the spacecraft, is tethers that can be reeled out from a spool or back in to change the unstretched length
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3.2.2 Hinges The other constrained structural configuration we shall consider is that in which two hinged panels are deployed by means of coilgenerated forces. For the Unfold configuration, coils are embedded or otherwise attached to rigid panels aligned on one edge and connected with a hinge, with motion limited to rotation around that hinge. This configuration is useful for the deployment of solar panels or electronically steered radar arrays or other segmented flat structures. When current is run through the coils, the force that they exert on each other is converted to torque around the hinge, causing them to rotate towards one another for two currents in the same direction and away from each other for two currents in the opposite direction, as in Figure 13.
Figure 13: Two coils repelling each other across a hinge in Unfold configuration
In order for a hinged structure to deploy and maintain a predetermined equilibrium position, there must be a restorative force that opposes the coil driven separation. In practice, this will be accomplished by incorporating a torque spring or springs into the hinge about which the panels rotate. Having a spring also supports Reconfiguration or Deformation of a structure operationally, since shutting off current through the coils allows the spring to pull the coils back together. Reconfiguration or Deformation is also possible by reversing the current in one coil in order to cause attraction between the coils. Let us consider the torques about the hinge more in detail. Let the angles of the two panels around the hinge axis be given by and . In the simplest case, if a torque spring is attached between the two panels, the spring torque is given by: ⃗
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where is the torque spring constant, ,   is the angle between the two panels, and is the natural angle of the spring. These angles are depicted in Figure 14, with the panels to which the coils are attached represented by lines from the bases of the coils to the hinge.
Figure 14: Hinge at equilibrium angle of spring and at stretched angle
Finally, in order to more effectively dampen any oscillations that occur about the equilibrium position, a rotational damping torque proportional to the angular velocity of the panels may be built into the hinge. In considering more fully the dynamics of the coils about a hinge, we first make some assumptions: 1) The system comprises two panels, rotating about a hinge and in planes parallel to one another, with coils attached to each panel 2) The hinge is fixed with respect to some inertial coordinate system 3) The hinge is parallel to a unit vector which we denote ̂ 4) The coil is rigidly connected to the panel 5) The coils and the panels are all rigid members (a brief discussion of what happens when the panels are not completely rigid may be found below)
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These assumptions allow us to directly formulate our equations of motion. Since the hinge is assumed fixed with respect to the inertial coordinate system, we may use the parallel axis theorem in determining the moment of inertia of the system (since there is only one degree of freedom per coil in this system, we do not need to calculate the entire moment of inertia matrix). Let the distance from the hinge axis to the center of each coil as mounted on the panel be given by , and let the radius of each coil be given by , i = 1,2. A coil rotating about an axis that passes through its diameter has a moment of inertia given by . The moment arm is given by , so that, by the parallel axis theorem, the moment of inertia of the coil about the hinge axis is given by:
To get the moment of inertia of a single coilpanel construct, we must add to this the moment of inertia of the panel. Suppose the panel is rigid, rectangular, very thin compared to its length, and of mass . If it has a length of ,
so that the moment of inertia
about the hinge for a single coilpanel construct is:
Again, the dynamics of the hinge deployment as specified allow for one degree of freedom of motion for each of the two panels, parameterized as the angle of each panel with respect to some axis. Then, Newton’s 2nd law can be written in angular form:
̂ ̂
∑⃗ ∑⃗
It remains to specify the torques acting on each panel. We will first consider the restorative torque applied to each panel counter to the moments due to electromagnetic force. These will be applied by torque springs about the hinge axis, and they may be implemented in one of two ways. 38
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As mentioned above, the first way in which the torque spring may be implemented is by fixing it to each of the panels, so that the spring torque is a function of the difference in angle between  . Supposing the spring has a spring constant given by κ and an the two panels, unstretched angle between its end connections of , the torque applied to each panel due to the spring will be given by:
⃗ ⃗
̂ ̂
} (Connected spring)
One consequence of this method of implementing a torque spring is that if there is any kind of jerking motion of either coil, then the coils will both acquire a net rotation, executing a motion in which the line bisecting the two panels has a nonzero angular rate about the hinge axis (as in Figure 14). This is undesirable when deploying the panels to precise orientations with respect to inertial space is required. Nevertheless, it is favorable from the point of view of minimizing the number of possible variables in the system and in that it directly ties the motions of the two panels to one another, if the deployment action requires so. As an alternative, we may consider the case in which each panel is attached to its own spring, each spring in turn connected to a structure that may be considered fixed. In this case, we have a spring constant for each panel, and , and an unstretched angle for each, and , so that the spring torques are given by
⃗
(
⃗
(
)̂
} )̂
The spring torques are now applied independently, but at the cost of having to determine additional parameters in order to implement a given equilibrium configuration. In this report, we will focus solely on implementing the first, connected, spring torques. As mentioned, we may also incorporate damping torques into the motion of the panels. How we do so, however, depends on the nature of the hinge itself. If the hinge is made up of two separate elements, each attached to a panel and rotating about an axle not connected to either, then the damping force will be proportional to the angular velocity of each panel independently about the axis:
39
⃗
̇ ̂
⃗
̇ ̂
}
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Electromagnetic deployment of space structures
NASA NIAC Phase I Final Report
where is the damping constant for the torque damping. If, however, the hinge is such that one element rotates about the other, then the damping torque will be a function of the difference in angular rates of each panel:
⃗
( ̇
̇ )̂
⃗
( ̇
̇ )̂
}
In this report we consider the hinge assembly to be made of separate hinge elements as in the first formulation. It remains to determine what the torques about the hinge axis are due to electromagnetic interactions between the coils. If we designate the meeting point of the panels and the hinge as the yaxis through the origin of the inertial coordinate system, then we have
⃗
∮ ⃗
⃗
Loop
⃗
∮ ⃗ Loop
∮ Loop
⃗
∮ Loop
⃗
⃗
⃗  ⃗
⃗
⃗ 
where ⃗ is the distance from the origin to a point on coil 1, ⃗ is the distance from the origin to a point on coil 2, and ⃗ ⃗ ⃗ . The resulting torque will be in a direction parallel to the hinge axis.
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Electromagnetic deployment of space structures
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3.3 Dynamic model implementation The equations of motion shown above were implemented and validated in MATLAB using a combination of builtin MATLAB functions, MathWorks’ Simulink tool, and a number of customwritten scripts. Generally, dynamics involving the full BiotSavart force law were computed in Simulink (which allows considerably for the incorporation of custom scripts), while those implementing the dipole force law were computed using the builtin ode45 solver. In the former case, the need to numerically compute the line integrals and resulting forces and torques about the coils necessitated the development of custom scripts, while in the latter case, analytical forms for the force and moment equations lent themselves to calculation using one of MATLAB’s robust ordinary differential equation solvers. 3.3.1 General solution algorithm The twocoil model was constructed for use in studying the actuation of circular, magnetic coils modeled as rigid, or unchanged from their circular state in the process of actuation. This model applies to the Unfold, and Separate deployment configurations and the Deform, Reconfigure, and Refocus operational configurations, all shown in Figure 15. Though this report focuses on twocoil systems, the model could be expanded to include three or more coils, which would enable study of the Inflate configuration.
Figure 15: Rigid coil functional configurations
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Below, we detail the steps taken in solution of the general equations of motion:
The model begins by defining the positions of φ evenly spaced points around each of two coils with respect to a global coordinate origin and assigns values for current ( ), wire crosssectional area, wire density per unit length ρ, radii of the coils ( ), and the moments of inertia of each ring ( ). At each timestep, the force and torque on each coil as a result of the magnetic field of the other coil must be calculated. This is done using a double forloop that calculates the force and torque on each point around Coil 1 from every point on Coil 2 (and vice versa), summing the contributions on one point from Coil 2 and then summing all the resultant forces and torques around Coil 1 for a final force and torque value on the coil at that timestep. The resultant forces and torques are multiplied after the loops end by the currents and appropriate constants to complete the BiotSavart equation and calculation of Laplace force. Accelerations of the centers of mass are calculated by dividing the resultant electromagnetic force ( ⃗ ),) by the masses of the coils. The positions of the coils are updated using the position and the velocity from the previous timestep. To determine the rotation on the coil about its center of mass, the torques (one for each coil) are projected onto the local axes of the coils. The angular velocities, quaternions and direction cosines matrices are calculated from the torques, and the orientation of the coils in space are updated accordingly. Plot new location and repeat in next time step.
3.3.2 Note on solution of stiff equations ⃗
is dependent on and the time is discretized by timesteps of varying duration in the tether model discussed above. We see from the formula for ⃗ that the larger k is, the force is for a certain . ⃗ and the electromagnetic force larger the restorative ⃗ ⃗ act simultaneously over the course of a timestep ; as ⃗ stretches the tether, ⃗ is opposing this motion. The formula for ⃗ when ≥ is very stiff when using most materials. The ⃗ acting to accelerate the coils apart over a timestep creates a that causes a ⃗ that is much larger than the ⃗ for all but the smallest s (and thus the smallest s are required for these two opposing forces to be of similar magnitudes and allow the model to converge to the approximation of continuous motion). A Simulink model utilizing the variabletimestep ode45 is used to reach the small s when ≥ and use larger s when < for shorter overall run times.
42
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3.4 Validation of numerical models Validation of the numerical methods employed in solving the dynamical equations of the coils are carried out by comparing results in which the force law is governed by the BiotSavart law with those in the dipole approximation. Validation of the numerical approximation can be done via an approximation of the coils as magnetic dipoles at farfield distances. The farfield as discussed in this report is defined as the case in which the diameter D of each coil under consideration is much less than the distance between every pair of coils in the network:
⃗ 
;
A more thorough discussion of what constitutes this limit follows below. While validation against an exact analytic solution is not possible in the near field, we demonstrate that both the behavior of the numerical model in the near field is consistent with its behavior in the farfield, and that the behavior of the numerical model in the farfield matches that of the dipole. Thus, the validation provides confidence in the accuracy of the model. 3.4.1 Analytic dipole model Now, the forces exerted by two magnetic dipoles on each other are given by Schweighart [20]: ⃗
[
⃗⃗⃗
⃗⃗⃗
⃗
⃗⃗⃗
⃗
⃗⃗⃗
⃗⃗⃗
⃗
⃗⃗⃗
where ⃗ is the vector distance between the centers of the coils, ⃗⃗⃗ and ⃗⃗⃗ are the magnetic moments of each coil, defined by: ⃗⃗⃗
̂
⃗⃗⃗
⃗ ( ⃗⃗⃗
⃗)
⃗]
is the magnetic constant, and
̂
being the number of turns in coil i, being the current, the area enclosed by the loop, and ̂ being the unit vector along the axis of the coil, oriented in the direction dictated by the righthand rule following the current around the loop. For two identical, coaxiallyaligned coils with planes parallel to one another, the dipole force equation reduces to:
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⃗
NASA NIAC Phase I Final Report
⃗ 
̂
with ̂ the unit vector in the direction of ⃗ towards the other coil. The above formula is multiplied by 1 if the currents are in opposite directions. 3.4.2 Sources of error in numerical approximation There are two sources of error that contribute to the difference in force calculations between the numerical model and the farfield dipole approximation: 1) the discretization of the coils into straight line segments rather than a continuous curve, and 2) the distance between the coils. The errors resulting from these two sources decrease to zero as the number of line segments in the discretization or the distance between coils increases to infinity. Discretization: When using the BiotSavart force law, the number of calculations to determine the resultant force and torque on each coil due to the other’s magnetic field is O( ), with being the number of differential elements around the coil, making fine discretization of curved coils (though more accurate to the curve of the coil) computationally expensive when computing the electromagnetic force. Distance: The numerical force falls off as ⁄ and the dipole force falls off as ⁄ , so we expect for very small ( ) values that the dipole force dominates, but then the dipole force is much greater than the numerical force right after r = 1 and then the difference between the two calculated forces goes to zero as . A quick visualization of this trend is shown in Figure 16.
Figure 16: Proportional difference between numerical and dipole approximations
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There is not a specific ratio of to at which the farfield approximation becomes valid; rather, one can determine what constitutes an acceptable percent error between the two. Figure 17 plots the initial force between a 1m diameter coil and a 0.98m diameter coil facing each other against the distance between them in both the numerical model and the dipole approximation, showing the convergence of the initial force values as the distance between the coils increases. The coils are slightly different sizes in order to avoid overlap of the coils in space for very small (